Season chicken cutlets with salt, black pepper, and garlic powder.
Organize a three-stage breading station: flour in the first tray, eggs beaten with 1 tablespoon of Dijon mustard in the second, and crushed pretzels in the third.
Dredge chicken in flour, submerge in the egg wash, and coat firmly with pretzel crumbs, pressing to ensure adhesion.
Heat vegetable oil in a large skillet over medium heat. Sear chicken for 4 to 5 minutes per side until the crust is golden brown and the internal temperature reaches 165°F (74°C). Transfer to a wire rack.
In a separate saucepan over medium-low heat, melt butter and whisk in heavy cream and the remaining 2 tablespoons of Dijon mustard.
Gradually whisk in the shredded cheddar cheese and cayenne pepper, stirring constantly until the sauce is emulsified and smooth.
Plate the crispy chicken and immediately ladle the warm mustard-cheddar sauce over the top.